Extreme weather events have large consequences, dominating the impact of climate on society, but are very difficult to characterize and predict, being exceptionally rare and pathological outliers in the spectrum of weather events. A rare event with a 100-year return period takes, on average, 100 years of simulation time to appear just once, let alone a statistically significant number of times. One can collect more statistics by running models at reduced resolution, but this comes at the cost of bias. High-fidelity models are needed to resolve the relevant dynamics. Furthermore, even if we had abundant data on extreme events, they make up a complex and diverse ensemble that is difficult to describe. Extremes come in different shapes, sizes, and magnitudes. Precursors and first causes are highly sought after for forecasting, but untangling these from background weather variability can raise thorny statistical issues. This thesis addresses both questions, by advancing two ideas: (1) Transition path theory, or TPT, as a mathematical framework to describe the statistical ensemble of rare events of a certain type; and (2) dynamical Galerkin approximation, or DGA, as a computational method to compute those important quantities. Both ideas emerged from the molecular dynamics community, and, I believe, have considerable potential for use by the climate modeling community. I demonstrate these ideas by way of example, on a hierarchy of models of one particular atmospheric phenomenon: sudden stratospheric warming (SSW), a rapid, large-scale disturbance in the stratosphere. SSW is an archetype of a complicated, extreme event that develops suddenly, often defying long-term prediction, and with disputed mechanisms. The TPT lens reveals some interesting features of SSW as a statistical ensemble, including its precursors, rate, and seasonal distribution. The hope is that this new tool will inspire fruitful investigation of many other kinds of atmospheric extremes.